Optimal post-warranty replacement policy for second-hand products considering failure-count and repair-time thresholds

This paper proposes a novel post-warranty replacement policy for repairable second-hand products (SHPs), in which replacement decisions are jointly determined by failure-count and repair-time thresholds. While most existing post-warranty maintenance models rely on a single replacement criterion, such as age, operating time, or failure count, the proposed policy incorporates both failure frequency and repair severity, thereby providing a more realistic representation of the deterioration process of repairable SHPs. The SHP is assumed to be sold with a non-renewing prorated warranty during which periodic imperfect preventive maintenance (PM) is provided by the dealer and minimal repair is carried out at each failure. After warranty expiration, the product owner performs minimal repairs at each failure until replacement is triggered either when the cumulative number of failures reaches a predetermined limit or when the repair time exceeds a specified threshold. Under a non-homogeneous Poisson process (NHPP) framework, analytical expressions are derived for the expected life-cycle length, expected total cost, and expected long-run cost rate. The optimization problem is formulated to determine the optimal failure-count threshold and repair-time threshold that minimize the expected cost rate. Analytical results establish the existence and uniqueness of the optimal solutions under reasonable conditions. Numerical examples based on Weibull failure distributions and exponential repair-time distributions demonstrate that the two decision variables are strongly interdependent and should therefore be optimized jointly. The results further show that higher repair costs lead to earlier replacement decisions and more restrictive repair-time thresholds, whereas the effects of product age and warranty characteristics are primarily reflected through the virtual age at warranty expiration. The proposed model extends the literature on post-warranty maintenance of second-hand products by introducing a replacement framework based on two interacting factors of failure count and repair time and provides practical decision support for economically efficient maintenance and replacement planning.